Absolute value equations & inequalities

Distance from zero

The absolute value of a number is its distance from zero on the number line. Tap a few numbers, some positive and some negative, and see how far each is from $0$.

Tap a number to measure its distance from 0.

Show every number 5 away from 0

$5$ and $-5$ are on opposite sides, but both are $5$ steps away. Distance doesn't care about direction, so it's never negative:

$|5| = 5 \qquad |-5| = 5 \qquad |0| = 0$

Absolute value equations

$|x| = 5$ asks: which numbers are $5$ away from zero? You just found them: there are two, one on each side.

$|x - 3|$ is the distance between $x$ and $3$. So $|x - 3| = 5$ asks: which numbers are $5$ away from $3$? Explore before you calculate.

Tap a number to measure its distance from 3.

Show every number 5 away from 3

The algebra says the same thing. The inside, $x - 3$, is either $5$ or $-5$, so split into two cases:

Isolate it first

Get the absolute value alone before you split.

How many solutions?

Once the absolute value is alone, look at the other side:

Absolute value inequalities

Which numbers are less than $3$ away from zero? Predict the shape of the graph, then test.

$|x| < 3$

Tap a number on the line to test it.

Test every pointShow the solution set

They're trapped between $-3$ and $3$: an and inequality, $-3 < x < 3$.

Now flip the question. Which numbers are more than $3$ away?

$|x| > 3$

Tap a number on the line to test it.

Test every pointShow the solution set

They're out past either end: an or inequality, $x < -3$ or $x > 3$.

The same idea works with any center:

That's every number within $3$ of $2$. And when the inside is more involved, the shape is still the same:

Key ideas

• Absolute value is distance from zero, and $|x - c|$ is the distance from $x$ to $c$.
• Isolate the absolute value, then split into two cases, one on each side.
• $|\ldots| < k$ becomes $-k < \ldots < k$, and $|\ldots| > k$ becomes $\ldots < -k$ or $\ldots > k$.